Physics 272: Advanced Honors Physics II
Spring 2018

HW4 due in class Monday, February 12, 2018
Reading: Purcell 3.1-7


Note: the solution for each problem and your name should be written on a separate page (sorry, trees!) to facilitate the grading. Please paperclip the pages together.
I have put the answers at the bottom of this page. You can use them to check your own answers -- if they don't agree, you know you need to go back and look for at least one mistake.


1. A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b, as in the left figure above). The shell carries no net charge.
(a) Find the surface charge density sigma at R, at a, and at b.
(b) Find the potential at the center, using infinity as the reference point.
(c) Now the outer surface is touched to a grounding wire, which drains off charge and lowers the potential to zero (same as at infinity). How do your answers to (a) and (b) change?

2. Two spherical cavities, of radii a and b, are hollowed out from the interior of a (neutral) conducting sphere of radius R (as in the right figure above). At the center of each cavity a point charge is placed -- call these charges qa and qb.
(a) Find the surface charge densities sigma-a, sigma-b and sigma-R.
(b) What is the field outside the conductor?
(c) What is the field within each cavity?
(d) What is the force on qa and qb?
(e) Which of these answers would change if a third charge qc were brought near the conductor?
 
3. **3.54
Dividing the surface charge

4. **3.60 A three-shell capacitor

5. **3.63 Capacitance coefficients for shells

6. *3.38 Two charges and a plane

7. **3.43 Images from three planes

8. **3.68 Maximum energy storage between cylinders

ANSWERS

Note that in some cases these are partial answers -- just enough for you to check that you are doing the problem right.
1. (a) At a, the surface charge density is -q/(4 pi a2); (c) the surface charge density at a is unchanged
2. (b) E has magnitude k|qa+qb|/r2, radially outward if qa+qb>0.
3. sigma1 = (8/13) sigma
4. Q1 = Q/4
5. C11 = 4 pi epsilon0 ab/(a-b); C12 = -
4 pi epsilon0 ab/(a-b)
6. y = (0.306)l
7. 0.210 Q2 /(4 pi epsilon0 d2)
8. U=pi  epsilon0 a2 E2 / (2e) where e is the base of the natural logarithm.